GATE 2022 ME (ME2) – Question 48

Engineering Mathematics · Linear Algebra: Matrix algebra, systems of linear equations, eigen values and eigen vectors · 2 marks · Numerical answer

If the sum and product of eigenvalues of a $2\times2$ real matrix $\begin{bmatrix}3&p\\p&q\end{bmatrix}$ are 4 and −1 respectively, then $|p|$ is ______ (in integer).

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Correct answer: 2

Explanation

For a $2\times2$ matrix, the sum of the eigenvalues is the trace and the product is the determinant.

**Step 1: trace.**
$$3+q=4\;\Rightarrow\;q=1 .$$

**Step 2: determinant.**
$$\det=3q-p^2=-1\;\Rightarrow\;3(1)-p^2=-1\;\Rightarrow\;p^2=4 .$$

So $p=\pm2$ and
$$|p|=\mathbf{2}.$$